SummaryA systematic discussion of partitioning as a tool for matrix inversion is presented, together with various methods and applications which have been of help in actual computations. New concepts are introduced, among them those of super-matrix and of square partitioning. The most usual type of partitioning, that into 2x2 sub-matrices, is discussed in detail, showing the orderly arrangement of the calculations in an auxiliary matrix. Further sections deal with matrices of the continuant type, and with special types of symmetry in the arrangement of the sub-matrices. The greatest advantage of the method of partitioning for the inversion of these types (as compared with the elimination method) lies in a considerable reduction in the number of arithmetical operations.
In a recent note Professor Duncan gives a method for finding the reciprocal of a matrix which is triply partitioned both horizontally and vertically. In spite of the apparent complication of the formulae, such a procedure may shorten the calculations in the case when the submatrices involved are especially simple (e.g. null or diagonal), or when their arrangement exhibits some symmetry. In Ref. 1, the submatrices a to j of the desired reciprocal are obtained by a series of operations summarised in equations (3) to (19); these operations consist altogether of 5 inversions of submatrices, 29 matrix multiplications of various orders and 15 matrix additions.
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